Research ArticleFinite Precision Logistic Map between Computational Efficiencyand Accuracy with Encryption Applications

Wafaa S Sayed1 Ahmed G Radwan12 Ahmed A Rezk2 and Hossam A H Fahmy3

1Engineering Mathematics and Physics Department Faculty of Engineering Cairo University Giza 12613 Egypt2Nanoelectronics Integrated Systems Center Nile University Cairo 12588 Egypt3Electronics and Communication Engineering Department Faculty of Engineering Cairo University Giza 12613 Egypt

Correspondence should be addressed to Wafaa S Sayed wafaassayedengcuedueg

Received 30 July 2016 Revised 8 October 2016 Accepted 4 December 2016 Published 12 February 2017

Academic Editor Alicia Cordero

Copyright copy 2017 Wafaa S Sayed et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Chaotic systems appear in many applications such as pseudo-random number generation text encryption and secure imagetransfer Numerical solutions of these systems using digital software or hardware inevitably deviate from the expected analyticalsolutions Chaotic orbits produced using finite precision systems do not exhibit the infinite period expected under the assumptionsof infinite simulation time and precision In this paper digital implementation of the generalized logisticmapwith signed parameteris considered We present a fixed-point hardware realization of a Pseudo-Random Number Generator using the logistic map thatexperiences a trade-off between computational efficiency and accuracy Several introduced factors such as the used precision theorder of execution of the operations parameter and initial point values affect the properties of the finite precisionmap For positiveand negative parameter cases the studied properties include bifurcation points output range maximum Lyapunov exponent andperiod length The performance of the finite precision logistic map is compared in the two cases A basic stream cipher system isrealized to evaluate the system performance for encryption applications for different bus sizes regarding the encryption key sizehardware requirements maximum clock frequency NIST and correlation histogram entropy and Mean Absolute Error analysesof encrypted images

1 Introduction

Chaos theory is a branch of mathematics which preciselydescribes many of the dynamical systems that exhibit unpre-dictable yet deterministic behavior Chaotic generators canbe classified into discrete time maps and continuous timedifferential equations The remarkable importance of chaoticiterated maps in both modeling and information processinginmanyfields explains the need for their hardware analog anddigital realizations for example [1ndash4] Digital realizationsare generally more immune to the imperfections of realelectronic systems and more secure due to the easiness ofencryption and exhibit greater noise immunity Moreoverthey are composed of digital circuit components which arecheap and easily produced on a single chip Since early 1990sa new class of Pseudo-RandomNumberGenerators (PRNGs)based on the digitization of chaotic maps has gained anincreasing interest Discrete time chaotic maps are easier to

implement on digital platforms and their generalized forms[5 6] could be fully utilized to fit multiple applicationsThis paper is concerned with one of the most famous one-dimensional discrete time chaotic maps the logistic map

The conventional logistic map is a quadratic nonlinearmap [7] given by

119909119899+1 = 119891 (119909119899 120582) = 120582119909119899 (1 minus 119909119899) (1)

where 119909119899 is the iterated variable of the map and 120582 is acontrol parameter Despite the simplicity of its mathematicalrelation that uses simple and computationally fast operatorsit is highly rich in information and indications that are veryuseful in the field of chaos theory and chaotic systems It hasalso found its way among other chaotic generators to manypractical applications such as biology physics chemistry[8ndash10] pseudo-random number generation [11] secure dataand image transfer techniques [12ndash17] with recent security

HindawiComplexityVolume 2017 Article ID 8692046 21 pageshttpsdoiorg10115520178692046

2 Complexity

0 05 1 15x

minus05

120582 gt 0120582 lt 0

1205824

1205824

f(x

)

(a)

xmaxminus = 15

1

05

0

xminminus = minus05

xmax+ = 1

x

120582

120582max = 432120582b1+ = 10120582b1minus = minus1120582min = minus2

(b)

Figure 1 (a) Graph of 119891(119909) = 120582119909(1 minus 119909) (b) Bifurcation diagram of 119891(119909) versus the control parameter 120582

reevaluation of some of them such as in [18] and financialmodeling [19 20]

Generalized logistic map with signed parameter in whichthe bifurcation diagram extends in both positive and negativecontrol parameter sides has been analyzed in [6] It has beenshown that period doubling route to chaos occurs for convexmaps as well as concave maps This is shown in Figure 1(a)for different values of the control parameter 120582 The output innegative control parameter side called mostly positive maphas a wider range that extends asymmetrically with alternat-ing sign as shown in Figure 1(b) with suggested applicationsin [6]Thekey-points of the bidirectional bifurcation diagramare illustrated in Figure 1(b) including bifurcation points andoutput ranges in both sides which may be listed as follows

(i) For 120582 gt 0 the first bifurcation point in the positiveside 1205821198871+ = 1 where the type of the first nonzerosolution is a fixed point In addition the upper boundon the output119909max+ = 1which takes place at120582max = 4

(ii) For 120582 lt 0 the first bifurcation point in the negativeside 1205821198871minus = minus1 where the type of the first nonzerosolution is period-2 In addition the lower and upperbounds on the output are 119909minminus = minus05 and 119909maxminus =15 respectively which take place at 120582min = minus2

Unseen behavior lies between the analytical study ofchaos and its digital representation where the effect ondynamical properties is inevitable Several methods of digi-tization could come to mind such as the following softwareimplementations in floating-point arithmetic formats sim-ulations in fixed-point formats hardware realizations eitherin ASIC or in FPGAs and other digital implementationsSoftware simulations of digital chaotic maps are criticized forbeing unsuitable for direct application to hardware FPGAswhich imply specific assumptions

In this paper a hardware oriented analysis of finite pre-cision logistic map using fixed-point arithmetic is presented

accompanied by a digital hardware implementation of PRNGThe basic operations constituting the map are executedindividually in a sequential manner with the truncationstep implemented between themThroughout our discussionfour factors which affect the finite precision logistic map areconsidered Two of them are explicit and not new which arethe control parameter 120582 and the initial point 1199090 howeverfinitude adds up to their known effectsThe other two factorsare introduced by the digital representation which are theprecision or bus size 119901 and the order of execution 119891(119909) Aslight perturbation in any of these factors can yield mas-sively different responses with varied properties in low andintermediate precisions The effect of varying precision onseveral properties of the generalized logistic map with signedparameter and the differences from the analytical modelare discussed comparing positive and negative parametercases A basic stream cipher system based on the PRNGis downloaded on FPGA and tested with text and imageencryption applications

The rest of this paper is organized as follows First digitalrepresentation of chaotic systems and a literature survey onprevious related works are presented in Section 2 ThenSection 3 discusses how the one-dimensional logistic mapcan be represented in digital hardware realizations such thatit can work for either positive or negative parameter casesThe assumptions required to simulate this representation insoftware environments are also discussed Different versionsof the map are proposed based on the order of execution ofthe operations constituting its expression In Section 4 two ofthese versions are chosen primarily to conduct various exper-iments and demonstrate results including the following thebifurcation diagram its key-points time series periodicity ofthe generated sequence and maximum Lyapunov exponent(MLE)The effects of varying precision initial condition andorder of execution on the type of solution are statisticallyanalyzed comparing positive and negative parameter cases

Complexity 3

Section 5 presents a hardware realization of a stream ciphersystem for text and image encryption applications based onthe conventional logistic map as a Pseudo-Random NumberGenerator The encryption performance speed and imple-mentation area are evaluated at different bus sizes NISTrandomness tests correlation histogram entropy and MeanAbsolute Error analyses of encrypted images are carried outfor different bus sizes Finally Section 6 summarizes thecontributions of this paper

2 Digital Representation of Chaotic Systems

In the field of chaos-based communication and for practicalconsiderations a design guide of a computationally efficientPRNG using chaotic systems with finite precision is neededThe reasonwhy chaos-based PRNGs are suggested frequentlywithout paying attention to the effect of finite precision couldbe owed to the 120573-shadowing lemmaThis lemma ensures thatthere exists an exact chaotic orbit close to the pseudoorbitwith only a small error [21] However there is an argumentwith strong evidences that this lemma cannot be applied todigital chaos Previous research efforts have attempted eitherto come up with a theoretical formulation of the problemand its consequences or to experimentally obtain results andanalyze them in the aimof acquiring full understanding of theproblem In this experimental approach several propertiesof chaotic systems are used as indicators that can be used touncover security weakness hidden inside some digital chaoticciphers

There are two aspects of regarding digitally implementedchaos either redefining the equation in a digital form andconfining outputs to the integer domain [22 23] or digitallyimplementing it in finite precision [24ndash30] Since we areconcerned with digital implementations a review of severalstudies about the effect of finite precision on the propertiesof chaotic systems is presentedThe problem of simulating orimplementing digital chaos is composed of two parts finitetime and finite precision Sentences like steady state or thelimit as the number of discrete time steps approaches infinityno longer carry the same meaning The behavior of a limitednumber of time samples can be recorded for some finiteprecision that is there is no practical implementation thatis equivalent to infinite time or precision

21 Continuous Time Chaotic Systems Corless in [24] dis-cussed numerical simulations with finite time and precisionof chaotic dynamical systems and how much they shouldbe trusted He suggested that the computed orbit and theaccompanied value for MLE could be falsely interpretedas chaotic (or nonchaotic) This could be owed to the ill-conditioned nature of chaotic dynamical systems wheresmall errors in initial conditions or involved operations areexponentially amplifiedwith time Consequently nomeasureexists of howmuch the actual response obtained shall deviatefrom the expected behavior and this deviation cannot betracked as time progresses

Several recent studies for the effect of limited precisionon the properties of digitally implemented continuous timechaotic systems have been conducted For example in [27

28] fully digital implementations of several 3rd-order ODE-based chaotic systems have been studied The thresholdminimum precision required for chaos has been decided tobe in the range of 8 to 11 fractional bits

It is expected in advance that simpler systems with lessdynamics such as one-dimensional discrete time logistic mapneeds a higher threshold minimum precision The variationin the response is expected to be slower with varying the usedprecision

22 Discrete Time Chaotic Systems Li et al in [25] studieddigitization of one-dimensional piece-wise linear chaoticmaps (PWLCM) and suggested several ways to reduce itsnegative effectThe effect of finite precision on the periodicityof a PRNG based on the logistic map has been explored in[26] The algorithm employs truncation in a single-precisionfloating-point environment after converting the binary32format to a denormalized binary fraction Truncation takesplace only after the execution of the whole expressionconsidering the subtraction followed by two multiplicationoperations in (1) as a single operation However this is notsuitable for a fixed-point arithmetic FPGA implementation

3 Fixed-Point Representation ofthe Logistic Map

Fixed-point representation uses integer hardware operationscontrolled by a given convention about the location of thefractional point Our discussion focuses on computationallyefficient fixed-point implementations of chaos specificallyone-dimensional logistic map The reason is that fixed-point arithmetic is significantly faster and less expensivethan an equivalent floating-point hardware implementationIn addition most commercial arithmetic logic unit (ALU)hardware for example FPGAs is based on it Such hardwarebuses typically offer between 8 and 64 bits of precision31 Assumptions of Fixed-Point Binary Representation Usingfinite precision fixed-point binary system the evaluation ofthe logistic map function is carried out in a similar manner toa microprocessor instruction set that is it is subdivided intoa sequence of basic operationsMATLAB fixed-point toolboxis used to simulate digital representation of the logistic mapon FPGA

The integer parts of the included ranges 120582 isin [minus2 4]and 119909 isin [minus05 15] are totally representable in 4 bits intworsquos complement coding These ranges correspond to thetwo maps positive logistic map and mostly positive logisticmap as illustrated before in Figure 1 It is guaranteed thatthe resulting value 119909 isin [minus05 15] is bounded that is noextra bits are needed for handling overflow conditions Thetotal number of bits or bus size is denoted by 119901 for precisionwhich is represented as 119901119894 integer bits and 119901119891 fractional bitssuch that 119901 = 119901119894 + 119901119891

Values of 119901 starting from a lower bound of 8 correspondto the least bus size offered by FPGAs that is 4 bits inboth the integer and the fractional parts A reasonable upperbound 119901 = 27 resembles the equality of the number offractional bits 119901119891 = 23 to the number of bits in the

4 Complexity

+ minus

1 xx120582

f1

times

times

+ minus

x 1 x120582

f2

times

times+ minus

x 1 x120582

f3

times

times

+ minus

xxx120582

f4

times

times

xxx 120582120582

f5

+ minus

timestimes

times

+ minus

xx120582

f6

timestimes

(a) f1(x 120582) = 120582(x(1 minus x)) (b) f2(x 120582) = (120582x)(1 minus x) (c) f3(x 120582) = (120582(1 minus x))x

(d) f4(x 120582) = (120582(x minus xx)) (e) f5(x 120582) = (120582x) minus (120582(xx)) (f) f6(x 120582) = (120582x) minus ((120582x)x)

Figure 2 Six different maps in fixed-point arithmetic

fractional part 119891 of the single-precision binary floating-pointrepresentation which is the smallest precision used in mostsoftware implementations Such upper bound is tentative andchanges frequently in the rest of the paper according to thesensitivity of the studied property to precisionThe followingassumptions are made in simulating digital representation ofthe logistic map

(i) The values of 119909 and 120582 and the output of each basicoperation are fixed-point variables stored in finitelength Registers

(ii) All operations are carried out assuming truncation(iii) Tworsquos complement coding representation is used

32 Different Maps in Fixed-Point Arithmetic The result ofthe studied function given by (1) can be calculated inmultipleways in fixed-point arithmetic For instance consider theexpressions shown in Figures 2(a) 2(b) and 2(c) Arethey equivalent Is the associativity property maintained ina fixed-point system Moreover the operations could begrouped such that suboperations are performed in differentorder as in Figures 2(d) 2(e) and 2(f) The six shownalternatives have been chosen to be consideredThe hardwareresources needed for executing each of them are also illus-trated in the form of Registers and arithmetic units (addersmultipliers etc)

Throughout the rest of the discussion all results areobtained by MATLAB starting at initial point 1199090 = 05discarding the first 1000 iterations and considering the next500 ones except where stated otherwise Figure 3 showsthe bifurcation diagram versus the control parameter 120582 at119901 = 9 which is expected to exhibit different phases ofbehavior as those shown in Figure 1(b) However it couldbe noticed that different orders of execution yield bifurcationdiagrams that are different from those expected regarding thefollowing key-points output range transition between typesof responses and density of points at ranges that are supposedto exhibit chaotic behavior

From the bifurcation diagrams 1198913(119909) and 1198916(119909) exhibitsmoother maps and a more similar behavior to that expectedfrom the logistic map compared to the other studied alterna-tivesHencemost of the following discussion concentrates onthese two versions of the logistic map the different propertiesthat they exhibit and howmuch they conform to the behaviorexpected from the mathematical analysis of the map

Various properties that have been considered as facts inmathematical analysis of the one-dimensional discrete timelogistic map are violated in finite precision environmentsSeveral examples are detailed below

(i) Initial points with a ldquo1rdquo in the least significant bit onlythat is 1199090 = 2minus119901119891 could cause the response to dieout In addition any quantity less than 2minus119901119891 will beconsidered zero after truncation to 119901119891 fractional bits

(ii) Analytically it would be expected from (1) that twoinitial points with difference = 1 yield the sameposttransient behavior since they have the same orbitHowever this property is not always satisfied in finiteprecision case due to truncation effects

(iii) The details of the bifurcation diagram over the wholerange of 120582 might slightly alter starting at differentinitial points Figure 4 shows that the bifurcationdiagram differs from the previous case starting atdifferent initial point 1199090 = 0125

4 Properties of the Selected Maps

In order to study the effect of increasing the number offractional bits and the impact of low precisions on the prop-erties of logistic map the bifurcation diagram of 1198913(119909) versusthe control parameter 120582 is plotted for different precisions inFigure 5 while that of1198916(119909) is shown in Figure 6 both startingat initial point 1199090 = 05 The resulting diagrams reveal thatthe properties of the logistic map are so much affected byprecision These properties include the following the key-points of the bifurcation diagram the number of levels orthe sequence of values at a fixed value of 120582 and the degree

Complexity 5

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 3 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 05 at 119901 = 9 (119901119891 = 5 bits)

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 4 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 0125 at 119901 = 9 (119901119891 = 5 bits)

of chaos or how much chaotic is the behavior at values of 120582near 120582max or 120582min

41 Key-Points of the Bifurcation Diagram The key-pointsof the bifurcation diagram play an important role as design

specifications of the logistic map utilized in various applica-tions specifically for generalized maps with extra parametersas those proposed in [5 6] and othersThese key-points differfor fixed-point representation from the analytical expectedbehavior defined in Section 1 due to truncation effects

6 Complexity

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 10

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 14

x

0 2 4minus2

120582

15

1

05

0

minus05

(d) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 119901 = 20

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 119901 = 24

Figure 5 Bifurcation diagram of 1198913(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 24

Figure 6 Bifurcation diagram of 1198916(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

However this difference decreases as 119901 increases as discussedbelow

411 Double-Precision Floating-Point Using double-precision floating-point calculations 1205821198871+ = 0969 119909max+ =0999738909465984 1205821198871minus = minus0967 119909minminus =minus049974730424707 and 119909maxminus = 15 These values for120582 are rounded to the third decimal digit after the pointand obtained comparing with epsilon machine (definedin MATLAB as 119890119901119904) The differences between the resultsof double-precision floating-point calculations and theexpected results could be owed to their relative inaccuracyThey do not satisfy neither infinite precision nor infinitetime conditions assumed analytically Hence floating-point

arithmetic implementations of chaotic generators and theirimpact on the various properties need to be studied as well

412 Fixed-Point Implementation Using fixed-point mapversions 1198913(119909) and 1198916(119909) let us consider the positive controlparameter side at first Figure 7(a) shows the values of 1205821198871+for 8 le 119901 le 26 for various initial points 1199090 = 0125 0250375 05 For the map 1198913(119909) Figure 7(a) shows that 1205821198871+starts at values higher than its analytic value ldquo1rdquo at lowprecisions and then starts to decrease gradually approachingldquo1rdquo on the other hand 1198916(119909) seems insensitive to precisionfrom the viewpoint of the value of1205821198871+ for these four values ofinitial point For output range calculations are performed atmultiple initial points and then the average of these different

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 3

Section 5 presents a hardware realization of a stream ciphersystem for text and image encryption applications based onthe conventional logistic map as a Pseudo-Random NumberGenerator The encryption performance speed and imple-mentation area are evaluated at different bus sizes NISTrandomness tests correlation histogram entropy and MeanAbsolute Error analyses of encrypted images are carried outfor different bus sizes Finally Section 6 summarizes thecontributions of this paper

2 Digital Representation of Chaotic Systems

In the field of chaos-based communication and for practicalconsiderations a design guide of a computationally efficientPRNG using chaotic systems with finite precision is neededThe reasonwhy chaos-based PRNGs are suggested frequentlywithout paying attention to the effect of finite precision couldbe owed to the 120573-shadowing lemmaThis lemma ensures thatthere exists an exact chaotic orbit close to the pseudoorbitwith only a small error [21] However there is an argumentwith strong evidences that this lemma cannot be applied todigital chaos Previous research efforts have attempted eitherto come up with a theoretical formulation of the problemand its consequences or to experimentally obtain results andanalyze them in the aimof acquiring full understanding of theproblem In this experimental approach several propertiesof chaotic systems are used as indicators that can be used touncover security weakness hidden inside some digital chaoticciphers

There are two aspects of regarding digitally implementedchaos either redefining the equation in a digital form andconfining outputs to the integer domain [22 23] or digitallyimplementing it in finite precision [24ndash30] Since we areconcerned with digital implementations a review of severalstudies about the effect of finite precision on the propertiesof chaotic systems is presentedThe problem of simulating orimplementing digital chaos is composed of two parts finitetime and finite precision Sentences like steady state or thelimit as the number of discrete time steps approaches infinityno longer carry the same meaning The behavior of a limitednumber of time samples can be recorded for some finiteprecision that is there is no practical implementation thatis equivalent to infinite time or precision

21 Continuous Time Chaotic Systems Corless in [24] dis-cussed numerical simulations with finite time and precisionof chaotic dynamical systems and how much they shouldbe trusted He suggested that the computed orbit and theaccompanied value for MLE could be falsely interpretedas chaotic (or nonchaotic) This could be owed to the ill-conditioned nature of chaotic dynamical systems wheresmall errors in initial conditions or involved operations areexponentially amplifiedwith time Consequently nomeasureexists of howmuch the actual response obtained shall deviatefrom the expected behavior and this deviation cannot betracked as time progresses

Several recent studies for the effect of limited precisionon the properties of digitally implemented continuous timechaotic systems have been conducted For example in [27

28] fully digital implementations of several 3rd-order ODE-based chaotic systems have been studied The thresholdminimum precision required for chaos has been decided tobe in the range of 8 to 11 fractional bits

It is expected in advance that simpler systems with lessdynamics such as one-dimensional discrete time logistic mapneeds a higher threshold minimum precision The variationin the response is expected to be slower with varying the usedprecision

22 Discrete Time Chaotic Systems Li et al in [25] studieddigitization of one-dimensional piece-wise linear chaoticmaps (PWLCM) and suggested several ways to reduce itsnegative effectThe effect of finite precision on the periodicityof a PRNG based on the logistic map has been explored in[26] The algorithm employs truncation in a single-precisionfloating-point environment after converting the binary32format to a denormalized binary fraction Truncation takesplace only after the execution of the whole expressionconsidering the subtraction followed by two multiplicationoperations in (1) as a single operation However this is notsuitable for a fixed-point arithmetic FPGA implementation

3 Fixed-Point Representation ofthe Logistic Map

Fixed-point representation uses integer hardware operationscontrolled by a given convention about the location of thefractional point Our discussion focuses on computationallyefficient fixed-point implementations of chaos specificallyone-dimensional logistic map The reason is that fixed-point arithmetic is significantly faster and less expensivethan an equivalent floating-point hardware implementationIn addition most commercial arithmetic logic unit (ALU)hardware for example FPGAs is based on it Such hardwarebuses typically offer between 8 and 64 bits of precision31 Assumptions of Fixed-Point Binary Representation Usingfinite precision fixed-point binary system the evaluation ofthe logistic map function is carried out in a similar manner toa microprocessor instruction set that is it is subdivided intoa sequence of basic operationsMATLAB fixed-point toolboxis used to simulate digital representation of the logistic mapon FPGA

The integer parts of the included ranges 120582 isin [minus2 4]and 119909 isin [minus05 15] are totally representable in 4 bits intworsquos complement coding These ranges correspond to thetwo maps positive logistic map and mostly positive logisticmap as illustrated before in Figure 1 It is guaranteed thatthe resulting value 119909 isin [minus05 15] is bounded that is noextra bits are needed for handling overflow conditions Thetotal number of bits or bus size is denoted by 119901 for precisionwhich is represented as 119901119894 integer bits and 119901119891 fractional bitssuch that 119901 = 119901119894 + 119901119891

Values of 119901 starting from a lower bound of 8 correspondto the least bus size offered by FPGAs that is 4 bits inboth the integer and the fractional parts A reasonable upperbound 119901 = 27 resembles the equality of the number offractional bits 119901119891 = 23 to the number of bits in the

4 Complexity

+ minus

1 xx120582

f1

times

times

+ minus

x 1 x120582

f2

times

times+ minus

x 1 x120582

f3

times

times

+ minus

xxx120582

f4

times

times

xxx 120582120582

f5

+ minus

timestimes

times

+ minus

xx120582

f6

timestimes

(a) f1(x 120582) = 120582(x(1 minus x)) (b) f2(x 120582) = (120582x)(1 minus x) (c) f3(x 120582) = (120582(1 minus x))x

(d) f4(x 120582) = (120582(x minus xx)) (e) f5(x 120582) = (120582x) minus (120582(xx)) (f) f6(x 120582) = (120582x) minus ((120582x)x)

Figure 2 Six different maps in fixed-point arithmetic

fractional part 119891 of the single-precision binary floating-pointrepresentation which is the smallest precision used in mostsoftware implementations Such upper bound is tentative andchanges frequently in the rest of the paper according to thesensitivity of the studied property to precisionThe followingassumptions are made in simulating digital representation ofthe logistic map

(i) The values of 119909 and 120582 and the output of each basicoperation are fixed-point variables stored in finitelength Registers

(ii) All operations are carried out assuming truncation(iii) Tworsquos complement coding representation is used

32 Different Maps in Fixed-Point Arithmetic The result ofthe studied function given by (1) can be calculated inmultipleways in fixed-point arithmetic For instance consider theexpressions shown in Figures 2(a) 2(b) and 2(c) Arethey equivalent Is the associativity property maintained ina fixed-point system Moreover the operations could begrouped such that suboperations are performed in differentorder as in Figures 2(d) 2(e) and 2(f) The six shownalternatives have been chosen to be consideredThe hardwareresources needed for executing each of them are also illus-trated in the form of Registers and arithmetic units (addersmultipliers etc)

Throughout the rest of the discussion all results areobtained by MATLAB starting at initial point 1199090 = 05discarding the first 1000 iterations and considering the next500 ones except where stated otherwise Figure 3 showsthe bifurcation diagram versus the control parameter 120582 at119901 = 9 which is expected to exhibit different phases ofbehavior as those shown in Figure 1(b) However it couldbe noticed that different orders of execution yield bifurcationdiagrams that are different from those expected regarding thefollowing key-points output range transition between typesof responses and density of points at ranges that are supposedto exhibit chaotic behavior

From the bifurcation diagrams 1198913(119909) and 1198916(119909) exhibitsmoother maps and a more similar behavior to that expectedfrom the logistic map compared to the other studied alterna-tivesHencemost of the following discussion concentrates onthese two versions of the logistic map the different propertiesthat they exhibit and howmuch they conform to the behaviorexpected from the mathematical analysis of the map

Various properties that have been considered as facts inmathematical analysis of the one-dimensional discrete timelogistic map are violated in finite precision environmentsSeveral examples are detailed below

(i) Initial points with a ldquo1rdquo in the least significant bit onlythat is 1199090 = 2minus119901119891 could cause the response to dieout In addition any quantity less than 2minus119901119891 will beconsidered zero after truncation to 119901119891 fractional bits

(ii) Analytically it would be expected from (1) that twoinitial points with difference = 1 yield the sameposttransient behavior since they have the same orbitHowever this property is not always satisfied in finiteprecision case due to truncation effects

(iii) The details of the bifurcation diagram over the wholerange of 120582 might slightly alter starting at differentinitial points Figure 4 shows that the bifurcationdiagram differs from the previous case starting atdifferent initial point 1199090 = 0125

4 Properties of the Selected Maps

In order to study the effect of increasing the number offractional bits and the impact of low precisions on the prop-erties of logistic map the bifurcation diagram of 1198913(119909) versusthe control parameter 120582 is plotted for different precisions inFigure 5 while that of1198916(119909) is shown in Figure 6 both startingat initial point 1199090 = 05 The resulting diagrams reveal thatthe properties of the logistic map are so much affected byprecision These properties include the following the key-points of the bifurcation diagram the number of levels orthe sequence of values at a fixed value of 120582 and the degree

Complexity 5

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 3 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 05 at 119901 = 9 (119901119891 = 5 bits)

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 4 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 0125 at 119901 = 9 (119901119891 = 5 bits)

of chaos or how much chaotic is the behavior at values of 120582near 120582max or 120582min

41 Key-Points of the Bifurcation Diagram The key-pointsof the bifurcation diagram play an important role as design

specifications of the logistic map utilized in various applica-tions specifically for generalized maps with extra parametersas those proposed in [5 6] and othersThese key-points differfor fixed-point representation from the analytical expectedbehavior defined in Section 1 due to truncation effects

6 Complexity

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 10

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 14

x

0 2 4minus2

120582

15

1

05

0

minus05

(d) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 119901 = 20

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 119901 = 24

Figure 5 Bifurcation diagram of 1198913(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 24

Figure 6 Bifurcation diagram of 1198916(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

However this difference decreases as 119901 increases as discussedbelow

411 Double-Precision Floating-Point Using double-precision floating-point calculations 1205821198871+ = 0969 119909max+ =0999738909465984 1205821198871minus = minus0967 119909minminus =minus049974730424707 and 119909maxminus = 15 These values for120582 are rounded to the third decimal digit after the pointand obtained comparing with epsilon machine (definedin MATLAB as 119890119901119904) The differences between the resultsof double-precision floating-point calculations and theexpected results could be owed to their relative inaccuracyThey do not satisfy neither infinite precision nor infinitetime conditions assumed analytically Hence floating-point

arithmetic implementations of chaotic generators and theirimpact on the various properties need to be studied as well

412 Fixed-Point Implementation Using fixed-point mapversions 1198913(119909) and 1198916(119909) let us consider the positive controlparameter side at first Figure 7(a) shows the values of 1205821198871+for 8 le 119901 le 26 for various initial points 1199090 = 0125 0250375 05 For the map 1198913(119909) Figure 7(a) shows that 1205821198871+starts at values higher than its analytic value ldquo1rdquo at lowprecisions and then starts to decrease gradually approachingldquo1rdquo on the other hand 1198916(119909) seems insensitive to precisionfrom the viewpoint of the value of1205821198871+ for these four values ofinitial point For output range calculations are performed atmultiple initial points and then the average of these different

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

4 Complexity

+ minus

1 xx120582

f1

times

times

+ minus

x 1 x120582

f2

times

times+ minus

x 1 x120582

f3

times

times

+ minus

xxx120582

f4

times

times

xxx 120582120582

f5

+ minus

timestimes

times

+ minus

xx120582

f6

timestimes

(a) f1(x 120582) = 120582(x(1 minus x)) (b) f2(x 120582) = (120582x)(1 minus x) (c) f3(x 120582) = (120582(1 minus x))x

(d) f4(x 120582) = (120582(x minus xx)) (e) f5(x 120582) = (120582x) minus (120582(xx)) (f) f6(x 120582) = (120582x) minus ((120582x)x)

Figure 2 Six different maps in fixed-point arithmetic

fractional part 119891 of the single-precision binary floating-pointrepresentation which is the smallest precision used in mostsoftware implementations Such upper bound is tentative andchanges frequently in the rest of the paper according to thesensitivity of the studied property to precisionThe followingassumptions are made in simulating digital representation ofthe logistic map

(i) The values of 119909 and 120582 and the output of each basicoperation are fixed-point variables stored in finitelength Registers

(ii) All operations are carried out assuming truncation(iii) Tworsquos complement coding representation is used

32 Different Maps in Fixed-Point Arithmetic The result ofthe studied function given by (1) can be calculated inmultipleways in fixed-point arithmetic For instance consider theexpressions shown in Figures 2(a) 2(b) and 2(c) Arethey equivalent Is the associativity property maintained ina fixed-point system Moreover the operations could begrouped such that suboperations are performed in differentorder as in Figures 2(d) 2(e) and 2(f) The six shownalternatives have been chosen to be consideredThe hardwareresources needed for executing each of them are also illus-trated in the form of Registers and arithmetic units (addersmultipliers etc)

Throughout the rest of the discussion all results areobtained by MATLAB starting at initial point 1199090 = 05discarding the first 1000 iterations and considering the next500 ones except where stated otherwise Figure 3 showsthe bifurcation diagram versus the control parameter 120582 at119901 = 9 which is expected to exhibit different phases ofbehavior as those shown in Figure 1(b) However it couldbe noticed that different orders of execution yield bifurcationdiagrams that are different from those expected regarding thefollowing key-points output range transition between typesof responses and density of points at ranges that are supposedto exhibit chaotic behavior

From the bifurcation diagrams 1198913(119909) and 1198916(119909) exhibitsmoother maps and a more similar behavior to that expectedfrom the logistic map compared to the other studied alterna-tivesHencemost of the following discussion concentrates onthese two versions of the logistic map the different propertiesthat they exhibit and howmuch they conform to the behaviorexpected from the mathematical analysis of the map

Various properties that have been considered as facts inmathematical analysis of the one-dimensional discrete timelogistic map are violated in finite precision environmentsSeveral examples are detailed below

(i) Initial points with a ldquo1rdquo in the least significant bit onlythat is 1199090 = 2minus119901119891 could cause the response to dieout In addition any quantity less than 2minus119901119891 will beconsidered zero after truncation to 119901119891 fractional bits

(ii) Analytically it would be expected from (1) that twoinitial points with difference = 1 yield the sameposttransient behavior since they have the same orbitHowever this property is not always satisfied in finiteprecision case due to truncation effects

(iii) The details of the bifurcation diagram over the wholerange of 120582 might slightly alter starting at differentinitial points Figure 4 shows that the bifurcationdiagram differs from the previous case starting atdifferent initial point 1199090 = 0125

4 Properties of the Selected Maps

In order to study the effect of increasing the number offractional bits and the impact of low precisions on the prop-erties of logistic map the bifurcation diagram of 1198913(119909) versusthe control parameter 120582 is plotted for different precisions inFigure 5 while that of1198916(119909) is shown in Figure 6 both startingat initial point 1199090 = 05 The resulting diagrams reveal thatthe properties of the logistic map are so much affected byprecision These properties include the following the key-points of the bifurcation diagram the number of levels orthe sequence of values at a fixed value of 120582 and the degree

Complexity 5

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 3 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 05 at 119901 = 9 (119901119891 = 5 bits)

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 4 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 0125 at 119901 = 9 (119901119891 = 5 bits)

of chaos or how much chaotic is the behavior at values of 120582near 120582max or 120582min

41 Key-Points of the Bifurcation Diagram The key-pointsof the bifurcation diagram play an important role as design

specifications of the logistic map utilized in various applica-tions specifically for generalized maps with extra parametersas those proposed in [5 6] and othersThese key-points differfor fixed-point representation from the analytical expectedbehavior defined in Section 1 due to truncation effects

6 Complexity

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 10

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 14

x

0 2 4minus2

120582

15

1

05

0

minus05

(d) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 119901 = 20

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 119901 = 24

Figure 5 Bifurcation diagram of 1198913(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 24

Figure 6 Bifurcation diagram of 1198916(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

However this difference decreases as 119901 increases as discussedbelow

411 Double-Precision Floating-Point Using double-precision floating-point calculations 1205821198871+ = 0969 119909max+ =0999738909465984 1205821198871minus = minus0967 119909minminus =minus049974730424707 and 119909maxminus = 15 These values for120582 are rounded to the third decimal digit after the pointand obtained comparing with epsilon machine (definedin MATLAB as 119890119901119904) The differences between the resultsof double-precision floating-point calculations and theexpected results could be owed to their relative inaccuracyThey do not satisfy neither infinite precision nor infinitetime conditions assumed analytically Hence floating-point

arithmetic implementations of chaotic generators and theirimpact on the various properties need to be studied as well

412 Fixed-Point Implementation Using fixed-point mapversions 1198913(119909) and 1198916(119909) let us consider the positive controlparameter side at first Figure 7(a) shows the values of 1205821198871+for 8 le 119901 le 26 for various initial points 1199090 = 0125 0250375 05 For the map 1198913(119909) Figure 7(a) shows that 1205821198871+starts at values higher than its analytic value ldquo1rdquo at lowprecisions and then starts to decrease gradually approachingldquo1rdquo on the other hand 1198916(119909) seems insensitive to precisionfrom the viewpoint of the value of1205821198871+ for these four values ofinitial point For output range calculations are performed atmultiple initial points and then the average of these different

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 5

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 3 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 05 at 119901 = 9 (119901119891 = 5 bits)

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 1198911(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 1198912(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 1198913(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(d) 1198914(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 1198915(119909)

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 1198916(119909)

Figure 4 Bifurcation diagram versus the control parameter 120582 for six different orders starting at 1199090 = 0125 at 119901 = 9 (119901119891 = 5 bits)

of chaos or how much chaotic is the behavior at values of 120582near 120582max or 120582min

41 Key-Points of the Bifurcation Diagram The key-pointsof the bifurcation diagram play an important role as design

specifications of the logistic map utilized in various applica-tions specifically for generalized maps with extra parametersas those proposed in [5 6] and othersThese key-points differfor fixed-point representation from the analytical expectedbehavior defined in Section 1 due to truncation effects

6 Complexity

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 10

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 14

x

0 2 4minus2

120582

15

1

05

0

minus05

(d) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 119901 = 20

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 119901 = 24

Figure 5 Bifurcation diagram of 1198913(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 24

Figure 6 Bifurcation diagram of 1198916(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

However this difference decreases as 119901 increases as discussedbelow

411 Double-Precision Floating-Point Using double-precision floating-point calculations 1205821198871+ = 0969 119909max+ =0999738909465984 1205821198871minus = minus0967 119909minminus =minus049974730424707 and 119909maxminus = 15 These values for120582 are rounded to the third decimal digit after the pointand obtained comparing with epsilon machine (definedin MATLAB as 119890119901119904) The differences between the resultsof double-precision floating-point calculations and theexpected results could be owed to their relative inaccuracyThey do not satisfy neither infinite precision nor infinitetime conditions assumed analytically Hence floating-point

arithmetic implementations of chaotic generators and theirimpact on the various properties need to be studied as well

412 Fixed-Point Implementation Using fixed-point mapversions 1198913(119909) and 1198916(119909) let us consider the positive controlparameter side at first Figure 7(a) shows the values of 1205821198871+for 8 le 119901 le 26 for various initial points 1199090 = 0125 0250375 05 For the map 1198913(119909) Figure 7(a) shows that 1205821198871+starts at values higher than its analytic value ldquo1rdquo at lowprecisions and then starts to decrease gradually approachingldquo1rdquo on the other hand 1198916(119909) seems insensitive to precisionfrom the viewpoint of the value of1205821198871+ for these four values ofinitial point For output range calculations are performed atmultiple initial points and then the average of these different

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

6 Complexity

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 10

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 14

x

0 2 4minus2

120582

15

1

05

0

minus05

(d) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(e) 119901 = 20

0 2 4minus2

120582

15

1

05x

0

minus05

(f) 119901 = 24

Figure 5 Bifurcation diagram of 1198913(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

0 2 4minus2

120582

15

1

05x

0

minus05

(a) 119901 = 12

0 2 4minus2

120582

15

1

05x

0

minus05

(b) 119901 = 18

0 2 4minus2

120582

15

1

05x

0

minus05

(c) 119901 = 24

Figure 6 Bifurcation diagram of 1198916(119909) versus the control parameter 120582 for different values of bus size starting at 1199090 = 05

However this difference decreases as 119901 increases as discussedbelow

411 Double-Precision Floating-Point Using double-precision floating-point calculations 1205821198871+ = 0969 119909max+ =0999738909465984 1205821198871minus = minus0967 119909minminus =minus049974730424707 and 119909maxminus = 15 These values for120582 are rounded to the third decimal digit after the pointand obtained comparing with epsilon machine (definedin MATLAB as 119890119901119904) The differences between the resultsof double-precision floating-point calculations and theexpected results could be owed to their relative inaccuracyThey do not satisfy neither infinite precision nor infinitetime conditions assumed analytically Hence floating-point

arithmetic implementations of chaotic generators and theirimpact on the various properties need to be studied as well

412 Fixed-Point Implementation Using fixed-point mapversions 1198913(119909) and 1198916(119909) let us consider the positive controlparameter side at first Figure 7(a) shows the values of 1205821198871+for 8 le 119901 le 26 for various initial points 1199090 = 0125 0250375 05 For the map 1198913(119909) Figure 7(a) shows that 1205821198871+starts at values higher than its analytic value ldquo1rdquo at lowprecisions and then starts to decrease gradually approachingldquo1rdquo on the other hand 1198916(119909) seems insensitive to precisionfrom the viewpoint of the value of1205821198871+ for these four values ofinitial point For output range calculations are performed atmultiple initial points and then the average of these different

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 7

11 14 17 20 23 268p

1

105

11

115

120582b1

+

f3(x)f6(x)

(a)

f3(x)f6(x)

092

094

096

098

1

xm

ax+

avg

11 14 17 20 23 268p

(b)

Figure 7 The key-points of the bifurcation diagram for the two positive control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average maximum value

results is considered Figure 7(b) shows the average value ofthe upper bound 119909max+avg on responses starting at multipleinitial conditions versus different precisions For both maps119909max+avg starts at values lower than its analytic value ldquo1rdquo atlow precisions then starts to increase gradually with somefluctuations approaching ldquo1rdquo

Similarly the key-points of the bifurcation diagram in thenegative control parameter side are studied Figure 8(a) showsthe values of 1205821198871minus where similar comments to the positivecontrol parameter case could describe the plot but for theabsolute value |1205821198871minus| instead Figure 8(b) shows the values of119909minminusavg at different precisions whereas Figure 8(c) shows thevalues of 119909maxminusavg The map 1198916(119909) seems to be less sensitive toprecision variation than 1198913(119909) The average minimum value119909minminusavg starts at low precisions with absolute values whichare lower than 05 whereas 119909maxminusavg starts with values whichare lower than 15 As precision increases the key-pointsapproach their analytical values

413 Sensitivity to Initial Conditions From Figures 7(a) and8(a) the values of 1205821198871+ and 1205821198871minus seem insensitive to the valueof initial point1199090 On the other hand the effect of initial pointon the values of 119909max+ 119909minminus and 119909maxminus cannot be avoided infinite precision implementations especially at low precisionsThe value 119909max+ occurs at 120582 = 4 whereas 119909minminus and 119909maxminusoccur at 120582 = minus2 Both values of 120582 exhibit maximum chaoticbehavior where sensitivity to initial conditions is a basiccharacteristic of the bifurcation diagram Analytically theldquoinfiniterdquo sequence generated at maximum chaotic behaviorhas lower and upper bounds which ldquomustrdquo be reached Yet infinite precision implementations the length of the generatedsequence is limited by both finite precision and timeThus itis not guaranteed whether its lower and upper bounds shall

coincide with the analytical values or not especially at lowprecisions

414 Precision Threshold By precision threshold we meanthe precision below which the properties of the map severelydeteriorate and above which changes become less significantIt does not mean that the behavior becomes exactly as theanalytical approach From Figures 7 and 8 it is clear that thevalues of the key-points derived through mathematical anal-ysis are the asymptotes that finite precision values approachas 119901 rarr infin The threshold minimum precision for key-pointscan be chosen as 119901 = 23 for 1198913(119909) and 119901 = 20 for 1198916(119909) forinstance

42 Time Series A chaotic posttransient response shouldhave a new value generated at each discrete time instant suchthat no periodicity can be recognized In this section the timeseries at values of120582 that are supposed to be chaotic are studiedin finite precision The effects of varying the used precisionand the initial point at which the orbit starts are explained

Values near 120582 = 4 or 120582 = minus2 which exhibit the widestchaotic response rich in applications are chosen Figures 9and 10 show the time series at 120582 = 39375 of the maps 1198913(119909)and 1198916(119909) respectively starting at different initial conditionsfor different precisions This specific value corresponds to anexactly representable fixed-point number with four fractionalbits corresponding to 119901 = 8 the narrowest precision thatis examined in our study Generally very low precisionsexhibit undesirable periodic behavior for almost all initialconditions and high precisions exhibit relatively long periodsfor some ormost of the initial conditions Time series at somecombinations of 119901 and 1199090 analytically expected to be chaoticare periodic instead for example Figure 10(b)

For example to study the reason behind the strangeresult obtained in Figure 10(b) the cobweb plot is shown

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

8 Complexity

minus115

minus11

minus105

minus1

120582b1

minus

11 14 17 20 23 268p

f3(x)f6(x)

(a)

11 14 17 20 23 268p

minus05

minus048

minus046

minus044

minus042

xm

inminus

avg

f3(x)f6(x)

(b)

11 14 17 20 23 268p

11

12

13

14

15

xm

axminus

avg

f3(x)f6(x)

(c)

Figure 8 The key-points of the bifurcation diagram for the two negative control parameter map versions 1198913(119909) and 1198916(119909) at differentprecisions (a) The first bifurcation point (b) The average minimum value (c) The average maximum value

in Figure 11 which is a rough plot for the orbit of 119909starting at different initial points 1199090 where the graph ofthe map function is sketched together with the diagonalline 119910 = 119909 Although the four cobweb plots seem differentthe posttransient solution colored in red and blue in caseof 1199090 = 0125 or 025 is the same as the plot in case of1199090 = 05 which fluctuates between six different values thatis period-6 The output states are as follows (these are thedecimal equivalents of the binary sequences representedin the used fixed-point representation at 119901 = 20 ie 16fractional bits) 05 rarr 0984375 rarr 00605621337890625rarr 02240142822265625 rarr 06844635009765625 rarr085040283203125 rarr 05009307861328125 rarr 0984375 Any orbit at the same 119901 and 120582 including one of thesesix values consequently the others will converge to thesame sequence of period-6 On the other hand the case

1199090 = 0375 exhibits a much longer sequence that could beconsidered ldquochaoticrdquo Assuming infinite precision period-6solution could be obtained through solving 1198916(119909119901) = 119909119901along with the stability analysis of the periodic point|(1198916)1015840(119909119901)| = 1 This would yield value(s) for 120582 at whichperiod-6 solution starts to appear and the correspondingvalues for 119909 According to [31] the value 120582 = 39375 is closeto one of these values

For further illustration Figure 12 shows the time series for120582 = 3984375 which is representable in 119901 ge 10 Although theneighborhood of this value does not contain near values thatgenerate periodic sequences some combinations of 119901 and1199090 could also yield faulty periodic response Figure 13 showsthe time series at 120582 = minus1984375 at different combinations of119891(119909) 119901 and 1199090 to illustrate the impact of finitude on mostlypositive logistic map too Calculations could be tracked

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 9

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 9 Time series of 1198913(119909) at 120582 = 39375 starting at different initial conditions and various precisions

similarly to illustrate how the recurrence settles to a specificperiodic sequence instead of the chaotic behavior expectedfrom mathematical analysis

Consequently the best expectation from the finite pre-cision logistic map is a pseudo-random sequence with longenough period The cases with significantly short periodsdepend on the combination of 120582 119901 and 1199090 which mightdrive us away from the expected chaotic behavior Thephenomenon of deviation from chaotic behavior does notappear in a continuous manner along with varying precisionThe solution could be chaotic at a certain precision119901 and thenbecome periodic at the next precision 119901 + 1 The same notecould be used to describe the case fixing the used precisionand varying the initial point Moreover all the studied ordersof execution are subjected to these truncation effects inlow precisions that yield output sequences with rather shortperiod

This section emphasizes the dependence of the type ofposttransient response on the initial conditions in low and

intermediate precisions The effects of using 119901 + 1 instead of119901 1199090 + 120575 instead of 1199090 (for small 120575) and any of the orders119891119894(119909) 119894 isin 1 2 3 4 5 6 are illustrated more in Section 43It is expected that at relatively high precisions 1199090 and 1199090 + 120575mostly yield the same type of posttransient solution exceptfor some cases in which truncation accidentally drives thesolution into a shorter periodThis is not to be confused withthe property of sensitivity to initial conditions and whetherthey are two different posttransient solutions or not which isdiscussed in [29]

From the previous discussion and visually inspectingfurther time series we could define the threshold minimumprecision according to the time series as the precision atwhich the response ldquoappearsrdquo chaotic for most of the initialpoints allowed by precision Values close to those suitable forbifurcation diagrams and key-points can be suggested

43 Periodicity of the Generated Sequence The sequencescorresponding to different parameters (119901 119891(119909) 120582 and 1199090)

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

10 Complexity

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(a) 119901 = 16

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(b) 119901 = 20

1

05

0

1

05

0

xx

nn

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

10000500001000050000

1

05

0

1

05

0

xx

nn

10000500001000050000

(c) 119901 = 24

Figure 10 Time series of 1198916(119909) at 120582 = 39375 starting at different initial conditions and various precisions

which are generated by finite precision logistic map donot follow an identified continuous manner as detailed inthe previous subsection It would be quite useful to pointout which combinations of the parameters yield responsesother than those expected through mathematical analysisReaching such combinations could be described as an attemptto solve the inverse problemwhere the value of 119909119899+1 is knownand the question is as follows what is the value of 119909119899 thathad yielded it for given values of 119901 and 120582 as well as anorder of execution 119891(119909) The answer to such a question is notstraightforward because successive values of 119909 are held in afinite lengthRegister Consequently the real number possiblyirrational yielded by solving the inverse problem should bemapped to finite precision fixed-point arithmetic Howeverthe nonlinearity of the relation representing the logistic mapmakes it hard to decide whether the mapped value should belower or higher than the analytical solution The number ofsteps above or below in the used precision cannot be easilydecided either

The type of posttransient response of the recurrence isobtained in terms of the length of the period formed by thesuccessive solutions A posttransient sequence of 119896 uniquevalues is described as ldquoperiod-119896rdquo which is affected by the fourfactors (119901 119891 120582 1199090) For example at 120582 = 4 and 119901 = 8 ourexperiments yield period lengths of 1 for the 15 available initialpoints using orders 1198911 1198914 1198915 and 1198916 However they yield thesame results as [26] with period lengths of either 1 or 3 at thesame corresponding initial points using 1198912 and 1198913 Moreoverlower values of120582whichwere not considered by [26] are foundout to yield longer period length of 4 at several combinationsof 120582 1199090 and 119891 emphasizing how the introduced factors affectthe period length

Figure 14(a) shows the maximum period or 119896 obtainedwith different orders of execution plotted versus precisionfor 120582 from 38125 to 4 in steps of 2minus119901119891 and all initial pointsallowed by the precision It seems that 1198913(119909) = (120582(1 minus119909))119909 yields relatively higher periods Some other orders areacceptable where the best are 1198912(119909) 1198913(119909) and 1198916(119909) as

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 11

0 05 10

02

04

06

08

1

xn+1

xn

(a)

0 05 10

02

04

06

08

1

xn+1

xn

(b)

0 05 10

02

04

06

08

1

xn+1

xn

(c)

0 05 10

02

04

06

08

1

xn+1

xn

(d)

Figure 11 Cobweb plots of 1198916(119909) at 120582 = 39375 and 119901 = 20 starting at different initial conditions (a) 1199090 = 0125 (b) 1199090 = 025 (c) 1199090 = 0375and (d) 1199090 = 05

08

06

1

04

02

x

n

1000050000

(a) 119901 = 13

08

06

1

04

02

x

n

10000500000

(b) 119901 = 16

x

n

1000050000

08

06

1

04

02

0

(c) 119901 = 19

x

n

1000050000

08

06

1

04

02

0

(d) 119901 = 22

Figure 12 Time series of 1198913(119909) at 120582 = 3984375 starting at 1199090 = 025 and various precisions

decided before from the bifurcation diagrams in Section 3It could be noticed that higher precisions provide morelevels among which the solution(s) could take their valuesallowing higher values for 119896 Figure 14(b) for 120582 from minus2 tominus18125 shows how the logistic map with negative controlparameter exhibits longer periods than that with positivecontrol parameter and at lower precisions In addition forselected combinations of the factors mostly positive mapexhibits more alternatives for unique values of 119896 with alonger maximum period for example 1198911 and 1198916 at 119901 = 11This indicates some merits of the logistic map with negativecontrol parameter over the conventional logistic map withpositive control parameter

It would be expected that the sequence period is governedby themaximumnumber of levels allowed by the chosen pre-cision for example 2119901119891 for 120582 gt 0 Yet this factor representsa rather loose upper bound imposed by the used precisionThe behavior of the studied map and whether its outputsequence covers all the allowed levels or not are anotherimportant point to consider which has been discussed in[26] For instance consider 119901 = 13 and the hypotheticalmaximum number of levels equal 29 = 512 levels Howeverexperimental results show that the maximum achievableperiod or number of distinct levels before repeating thesequence is roughly 50 for positive logistic map and 70 formostly positive logistic map Hence the dependence of the

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

12 Complexity

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(a) 119901 = 19 1198913(119909)

1

05

15

0

x

n

(A) x0 = 0125 (B) x0 = 0250

(C) x0 = 0375 (D) x0 = 0500

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

1

05

15

0

x

n

1000050000minus05

(b) 119901 = 19 1198916(119909)

Figure 13 Time series of mostly positive finite precision logistic map at 120582 = minus1984375 for different combinations of its parameters

8 9 10 11 12 13 147p

0

20

40

60

Max

per

iod

f1

f2

f3

f4

f5

f6

(a)

0

20

40

60

80

Max

per

iod

8 9 10 11 12 13 147p

f1

f2

f3

f4

f5

f6

(b)

Figure 14 Maximum period obtained with different orders of execution plotted versus precision 119901 = 8 rarr 13 for (a) 120582 = 38125 2minus119901119891 4and (b) 120582 = minus2 2minus119901119891 minus18125 and all initial points allowed by 119901

obtained period length on the used precision 119901 does notfollow a linear scaleThe efficiency of filling the allowed levelsis still insufficient in rather low and intermediate precisions

Figure 15 shows all unique periods obtained at a givenprecision for 1198913(119909) The legend shows the different colorscorresponding to a given period 119896 Figure 15 illustrates thatincreasing the used precision provides more combinationsof the parameter 120582 and the initial point 1199090 as expected Asprecision 119901 increases the number of alternatives for (119891 1199090 120582)increases exponentially and the statistics are limited by hugeprocessing time and memory requirements In addition

the number of obtained unique periods of output sequenceincreases with each bit of precision increased for low andintermediate precisions After reaching a high enough pre-cision threshold it is expected that the number of obtainedunique periods approaches settling Values close to theseobtained before in [29] may result from some combinationswhile other combinations may uncover surprising findings

44 Maximum Lyapunov Exponent Maximum Lyapunovexponent (MLE) is an indication whether the system exhibitschaotic behavior or not as itmeasures the rate of divergence of

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 13

0 05 138

385

39

395

4

120582

x0

1

2

3

6

9

(a) 119901 = 9 120582 = 38125 2minus5 4

1

2

19

22

3

5

4

6

7

8

9

10

13

12

14

16

0 05 138

385

39

395

4

120582

x08

9

10

13

12

14

16

(b) 119901 = 11 120582 = 38125 2minus7 4

minus2

minus195

minus19

minus185

minus18

1

2

35

6

7

120582

0 05 1 15minus05x0

(c) 119901 = 9 120582 = minus2 2minus5 minus18125

15161923

8

9

10

12

11

13

14

minus2

minus195

minus19

minus185

minus18

120582

0 05 1 15minus05x0

1

2

3

5

4

6

7

8

9

10

12

11

13

14

(d) 119901 = 11 120582 = minus2 2minus7 minus18125

Figure 15 All the generated period lengths in the studied ranges of parameter 120582 and all initial points allowed by 119901 for 1198913(119909) and differentprecisions

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

14 Complexity

Analytical derivative formulaForward difference formula

20 22 24 26 28 3018p

04

05

06

07

MLE

(a)

20 22 24 26 28 3018p

055

06

065

07

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 16 MLE evaluation in two different methods using 1198913(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

nearby initial points Two suggested methods for calculatingMLE in finite precision are explained in the next twosubsections

441 Analytical Derivative Formula MLE for discrete timemaps [7] is given by

MLE = lim119899rarrinfin

(1119899119899minus1sum119894=0

ln 100381610038161003816100381610038161198911015840 (119909119894)10038161003816100381610038161003816) (2)

where ln is the natural logarithm and 1198911015840(119909119894) is the firstderivative of the map equation with respect to 119909 at 119909 = 119909119894in addition to choosing 119899 large enough for example 119899 =50000 forMLE value to reach its steady state In the analyticalderivative formula we calculate MLE of the finite precisionlogistic map as follows

MLE = 1119899119899minus1sum119894=0

ln 1003816100381610038161003816120582 (1 minus 2119909119894)1003816100381610038161003816 (3)

442 Numerical Approximation of First Derivative The fol-lowing discussion attempts to investigate whether the numer-ical approximations for first derivative are more compatiblewith a discrete map than continuous differentiation rulesyielding 1198911015840(119909119894) = 120582(1 minus 2119909119894) Consider the forward differenceapproximation of the first derivative

1198911015840 (119909119894) = 119891 (119909119894 + Δ) minus 119891 (119909119894)Δ (4)

Substituting (4) in (2) and using the properties of logarithmwe get

MLE = 1119899119899minus1sum119894=0

ln1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816Δ (5a)

MLE = [1119899119899minus1sum119894=0

ln 1003816100381610038161003816119891 (119909119894 + Δ) minus 119891 (119909119894)1003816100381610038161003816] minus lnΔ (5b)

where the minimum value for Δ theoretically equals 2minus119901119891which is the minimum representable value However forpractical issues and to avoid overflow Δ is set to a quitehigher value Both (5a) and (5b) were found to be equivalentin double-precision floating-point and match the analyticalmethod for relatively high precisions Applying the one-dimensional time series toMLE calculation tools such as [32]used in [27 28] can be considered as a third approach

A threshold minimum precision of 119901 ge 22 from theviewpoint of MLE calculation can be suggested as illustratedby Figures 16 and 17 However results obtained in lowprecisions make us doubt the validity of both approaches forcalculatingMLE in rather low precisions and the reliability ofthe numbers computed through them in deciding how muchchaotic a system is However the problems in calculatingMLE using both methods are barely noticed at quite higherprecisions and dominate only at relatively low precisions 119901 lt22 as previously mentioned

We conclude that at rather low precisions MLE cannotbe used as a standalone indicator of chaotic behavior withoutconsidering the corresponding sequence length The value ofMLE calculated through either of the two methods could befalsely positive while the output sequence is clearly periodicThis result comes in accordancewith the discussion presentedin [24 25]

5 Encryption Applications

A Pseudo-Random Number Generator (PRNG) is realizedusing the conventional logisticmap (1) as shown in Figure 18The inputs of the system are the clock Reset 120582 and 1199090 Thearithmetic unit is used to compute 119909119899+1 using 119909119899 and 120582 TheRegister is used to provide 119909119899 to the arithmetic unitThe valueof 119909119899 is updated with Multiplexerrsquos output every clock cycle(iteration) In case of Reset (Reset = 1) the Multiplexer willprovide 1199090 to the Register Otherwise the Multiplexer willpass 119909119899+1 to the Register This unit is realized using VHDL

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 15

05

06

065

07M

LE

20 22 24 26 28 3018p

Analytical derivative formulaForward difference formula

(a)

20 22 24 26 28 3018p

058

06

062

064

066

MLE

Analytical derivative formulaForward difference formula

(b)

Figure 17 MLE evaluation in two different methods using 1198916(119909) starting at 1199090 = 0125 at (a) 120582 = 3984375 and (b) 120582 = minus1984375

1

D

Q

clk

1

0

Reset

Arithmetic unit

x0xn

xn+1 = f(xn 120582)

120582

times

times

minus+

Figure 18 Hardware realization of Pseudo-Random Number Gen-erator

and synthesized on Xilinx using XC5VLX50T as the targetFPGA Table 1 shows the hardware implementation area andthe maximum clock frequency for different bus sizes

A basic stream cipher system is realized as shown inFigure 19 The ciphering process is based on Xoring theinput data with a stream of random numbers The PRNGpresented in Figure 18 is used to output the randomnumberswhere 120582 and 1199090 are used as the encryption key Hencethe keyspace is 22119901 Every clock cycle the system generatesa new random number (RN) captures a new byte fromthe input data and Xores the 8 least significant bits of thegenerated RN with the input byte The system is realizedusing VHDL and synthesized on Xilinx using XC5VLX50TTable 2 summarizes the synthesis results for different bussizes The systemrsquos performance regarding the speed andimplementation area improves by decreasing the size of thebus Unfortunately decreasing the size of the bus leads to areduction in the size of the encryption key which makes thesystem vulnerable to brute force attacks Thus the size of thebus must be adjusted to satisfy the security level speed andhardware resources

PRNG

clk

Reset

XORInput Output

xn (7 0)

(7 0)(7 0)

x0 (p minus 1 0)

120582 (p minus 1 0)

Figure 19 Stream cipher system for encryption applications

The stream cipher system is downloaded on FPGA andtested with a text encryption applicationThe system displaysthe input word on the first line of the Kitrsquos LCD screenencrypts the word using the input key and displays theencrypted word on the second line of the screen Figure 20shows the results of the prototype for different bus sizesand encryption keys The system succeeds in encrypting theinput word (Hello World 2016) for all test cases Howeverin Figures 20(b) and 20(d) there is a relation between theinput text and the ciphered text which leads to poor encryp-tion In (b) the characters ldquolrdquo ldquospacerdquo and ldquoordquo are alwaystransformed to ldquodrdquo ldquo( rdquo and ldquogrdquo respectively Similarly in (d)the characters ldquolrdquo ldquospacerdquo and ldquoordquo are always transformed toldquo rdquo ldquo1rdquo and ldquorarrrdquo respectively This is due to the fact that for120582 = 2 the solution of the logistic map does not bifurcate asillustrated previously in Figure 3 Thus in order to improvethe encryption performance the values of 120582must be close to4

Moreover Table 3 shows the encryption results for afamous quote by Mahatma Gandhi [33] using 27-bit bussize and two different keys (11990901 1205821) = (0507874131237501969337) and (11990902 1205822) = (0757874131237501969337) The number of different bits between

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

16 Complexity

Table 1 Synthesis results for the logistic map using different bus sizes

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 21 2 104716MHz14 14 29 2 104251MHz18 18 37 2 103873MHz20 20 183 2 79590MHz27 27 253 4 61812MHz

Table 2 Synthesis results of the stream cipher system

Bus size Slice Registers (28800) Slice LUTs (28800) DSP48Es (48) Maximum clock frequency10 10 31 2 104716MHz14 14 37 2 104251MHz18 18 45 2 103873MHz20 20 191 2 79590MHz27 27 261 4 61812MHz

Table 3 Encryption results for a famous quote using a bus size of 27 bits and two different keys

Plain text Live as if you were to die tomorrow Learn as if you were to live forever Ciphertext (11990901 1205821)Ciphertext (11990902 1205822)

Bus size = 8 bits120582 = 375x0 = 05

(a)

Bus size = 8 bits120582 = 2

x0 = 05

(b)

Bus size = 27 bits 120582 = 3750008464x0 = 0500061512

(c)

Bus size = 27 bits

x0 = 0500061512120582 = 2000008464

(d)

Figure 20 Results of the prototype for different bus sizes and encryption keys

these two keys is reduced to one bit only in order to test thesystemrsquos key sensitivity The results show that the encryptedsentences are totally different which imply that the keysensitivity is high

The randomness of the PRNG is usually tested by thestandard NIST suite [34] in order to ensure high securityperformance Accordingly a VHDL test bench has beenimplemented to interface the Xilinx with the NIST suiteThistest bench is used to generate a text file which includes the8 LSBs of 125000 successive iterations The PRNG must passall the NIST tests in order to be considered as true randomsourceThe results of theNIST test for 8-bit bus size up to 45-bit bus size are summarized in Table 4 For all tests the initialpoint 1199090 is set to 05 + 2minus13 while 120582 is set to 4 minus 2minus119901119891 where

119901119891 is the number of fractional bits The threshold bus size forpassing all theNIST tests is 45 bits As the bus size is decreasedbelow this threshold the number of reported failures startsto increase until the PRNG fails in all the tests for 8-bit bussizeThe system succeeds in all the NIST tests for any bus sizegreater than 45 bits

Moreover the system has been tested with an imageencryption application as shown in Table 5The standard graycolor Lena (256 times 256) has been supplied to the streamcipher system in Figure 19Three bus sizes 11 bits 12 bits and45 bits have been used in the analysis

The vertical horizontal and diagonal pixel correlationcoefficients 120588 are calculated using (6a) (6b) and (6c) where119873 is the total number of pixels selected from the image and

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 17

Table4NISTresults

ford

ifferentb

ussiz

es

NISTtest(sam

ple1000

000bits

inleng

th)

8bits

27bits

34bits

36bits

38bits

40bits

42bits

45bitsrsquo119875va

lue

Syste

mparameters(120582

119909 0)(4minus

2minus4 05)

(4minus2minus2305

+2minus13)(4

minus2minus3005

+2minus13)(4

minus2minus3205

+2minus13)(4

minus2minus3405

+2minus13)(4

minus2minus3605

+2minus13)(4

minus2minus3805

+2minus13)(4

minus2minus4105

+2minus13)

Frequency

MM

M

0788699

Blockfre

quency(119898=

128)M

M

0880935

Cusum-Forward

MM

M

032118

3

Cusum-Reverse

MM

M

0511427

Runs

MM

0950620

Long

runs

ofon

eM

MM

0301448

Rank

MM

0178158

SpectralDFT

MM

MM

M

0581909

Non

overlapp

ingtemplates

MM

MM

MM

M064

5372

Overla

ppingtemplates(119898=

9)M

MM

0566886

Universal

M

0725132

Approxim

atee

ntropy(119898=

10)M

MM

MM

0877618

Rand

omexcursions

MM

MM

MM

M0970335

Rand

omexcursions

varia

ntM

MM

MM

M

0125786

Linear

complexity(119872

=500)

M

0113062

Seria

l(119898=16

)M

MM

MM

0115512

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

18 Complexity

Table5Im

agee

ncryptionanalysis

Analysis

Lena256

times256

EncryptedIm

age11-b

itsEn

cryptedIm

age12-bits

EncryptedIm

age4

5-bits

Keyspace

NA

222224

290Ke

yparameters

NA

120582=4minus2minus7

120582=4minus2minus8

120582=4minus2minus41

119909 0=05

119909 0=05

119909 0=05+

2minus13

Image

Verticalcorrelation

09693

06565

08988

00037

Horizon

talcorrelation

09400

064

82000

43000

073799

Diagonalcorrelation

09276

05858

minus00027

minus00051

Histogram119910-ax

isob

served

coun

t

1112131415161718191

101111121131141151161171181191201211221231241251

0

100

200

300

400

500

600

050100

150

200

250

300

350

400

450

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

350

400

1112131415161718191

101111121131141151161171181191201211221231241251

050100

150

200

250

300

1112131415161718191

101111121131141151161171181191201211221231241251

119909-axis

colorintensity

Entro

pyNA

78737

79823

79973

MAE

NA

413697

688534

778117

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 19

Table 6 Synthesis results of the 45-bit stream cipher system

Resource Stream cipher system Stream cipher system with HDMISlice Registers (28800) 45 278Slice LUTs (28800) 326 936DSP48Es (48) 20 20Maximum clock frequency 43530MHz 43513MHz

(a) (b)

Figure 21 Standalone image encryption system (a) Decrypted image (b) Encrypted image

(119909(119894 119895) and 119910(119894 119895)) are two adjacent pixels For highly securedimage 120588must be very close to zero

cov (119909 119910) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)(119910119894 minus 1119873119873sum119895=1

119910119895) (6a)

119863 (119909) = 1119873119873sum119894=1

(119909119894 minus 1119873119873sum119895=1

119909119895)2

(6b)

120588 = cov (119909 119910)radic(119863 (119909))radic(119863 (119910)) (6c)

The histogram analysis is used to observe the distributionof the pixelrsquos color intensity For highly secured image theobserved count for each of 28 color levels must be the sameover the entire image

The entropy of the ciphered image is calculated using (7)where 119901(119904119894) is the probability of symbol 119904119894 119901(119904119894) is computedby dividing the observed count of 119904119894 in the ciphered image bythe size of the image For highly secured image the entropymust be close to 8

Entropy = minus 28sum119894=1

119901 (119904119894) log2119901 (119904119894) (7)

The Mean Absolute Error (MAE) between the originalimage 119875 and the ciphered image 119862 is calculated using (8)where 119894 and 119895 are the pixel indices and119872 and119873 are the widthand height of the image respectively

MAE = 1119872 times119873119873sum119894=1

119872sum119895=1

1003816100381610038161003816119875 (119894 119895) minus 1198621 (119894 119895)1003816100381610038161003816 (8)

The encryption analysis in Table 5 shows poor resultsfor the 11-bit bus size Fortunately as the bus size increases120588 coefficient approaches 0 the pixelsrsquo intensity distributionbecomes flat entropy approaches 8 and MAE increases Thebest encryption result is achieved by setting the bus size to 45bits which is identified previously in the NIST analysis

Table 6 shows synthesis results of the 45-bit stream ciphersystem in addition to standalone encryption system thatdisplays the encrypted and decrypted image on a screenthrough an HDMI connector The ldquoLenardquo image is stored inthe internal RAM of the FPGA Figure 21 shows the proposedstandalone image encryption system

The proposed stream cipher system with 8-bit up to 27-bit bus size was acceptable in the text encryption applicationdue to the small amount of input data However theNIST andimage encryption analysis have showed that it is necessaryto extend the bus size to 45 bits in order to successfullyencrypt large amount of data The performance can be fur-ther enhanced through feedback delay elements nonlinearblocks and permutation stages as well as postprocessingtechniques to get better results for security and sensitivityanalyses

6 Conclusions

In this paper a methodology for fixed-point simulation ofthe generalized one-dimensional logistic map with signedparameter and an equivalent hardware realization werepresented Criteria for choosing the bus size for digitallyimplemented chaotic systems and specifying the number ofinteger bits were suggested Various numerical simulationsfor the properties of the finite precisionmap over the availableprecisions orders of execution parameter and initial point

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

20 Complexity

were carried out in comparison with those obtained throughmathematically analyzing the map over the infinite realfield These factors were shown to affect several propertiesincluding the following the bifurcation diagram its key-points periodicity of the generated sequence and maxi-mum Lyapunov exponent for positive and negative controlparameter casesMoreover a hardware realization of a streamcipher system for text and image encryption applicationsbased on the conventional logistic map as a Pseudo-RandomNumber Generator was presented Encryption results showthe trade-off between security level on the one hand andspeed and hardware resources on the other hand The longerperiod lengths provided by mostly positive map suggest itsuse as one remedy for the problem of short cycles of finiteprecision chaotic maps compared with other PRNGs It canbe an additional enhancement beside using higher finiteprecisions cascading multiple chaotic systems and usingperturbation based systems for either the control param-eter or the iterated variable with different configurationsScaling factors can further provide wider output ranges andhence increase period lengths Moreover we recommendthat future implementations of chaotic systems on digitalplatforms consider the listed factors and their effects onvarious properties and then provide their implementationdetails for reproducibility For example we have shown thatdifferent orders of execution and parameter values yielddifferent period lengths For finite precision systems we donot guarantee that 120582 = 4 or which order of execution yieldsthe longest period Hence the different alternatives needto be considered similar to how initial conditions effect isconsidered The procedure presented in this paper can becarried out for other generalized versions of the logistic mapand other chaotic systems according to the allowed rangesof different parameters in order to study the impact of finiteprecision fixed-point implementation on their properties

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] K M Cuomo and A V Oppenheim ldquoCircuit implementationof synchronized chaos with applications to communicationsrdquoPhysical Review Letters vol 71 no 1 pp 65ndash68 1993

[2] S Mandal and S Banerjee ldquoAnalysis and CMOS implementa-tion of a chaos-based communication systemrdquo IEEE Transac-tions on Circuits and Systems I Regular Papers vol 51 no 9 pp1708ndash1722 2004

[3] A Radwan A Soliman and A S Elwakil ldquo1-D digitally-controlled multiscroll chaos generatorrdquo International Journal ofBifurcation and Chaos vol 17 no 1 pp 227ndash242 2007

[4] M A Zidan A G Radwan and K N Salama ldquoControllable V-shape multiscroll butterfly attractor system and circuit imple-mentationrdquo International Journal of Bifurcation and Chaos vol22 no 6 Article ID 1250143 2012

[5] S K Abd-El-Hafiz A G Radwan and S H AbdEl-HaleemldquoEncryption applications of a generalized chaotic maprdquo Applied

Mathematics amp Information Sciences vol 9 no 6 pp 3215ndash32332015

[6] W S Sayed A G Radwan and H A H Fahmy ldquoDesignof positive negative and alternating sign generalized logisticmapsrdquo Discrete Dynamics in Nature and Society vol 2015Article ID 586783 23 pages 2015

[7] K T Alligood T D Sauer and J A Yorke ChaosmdashAnIntroduction to Dynamical Systems Textbooks in MathematicalSciences Springer-Verlag New York NY USA 1997

[8] E Scholl Nonlinear Spatio-Temporal Dynamics and Chaos inSemiconductors vol 10 Cambridge University Press 2001

[9] S H Strogatz Nonlinear Dynamics and Chaos With Applica-tions to Physics Biology Chemistry and Engineering WestviewPress 2014

[10] R K Upadhyay and P Roy ldquoDisease spread and its effect onpopulation dynamics in heterogeneous environmentrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 26 no 1 Article ID 1650004 27 pages 2016

[11] L Liu and S Miao ldquoThe complexity of binary sequences usinglogistic chaotic mapsrdquo Complexity vol 21 no 6 pp 121ndash1292016

[12] S C Phatak and S S Rao ldquoLogistic map a possible random-number generatorrdquo Physical Review E vol 51 no 4 pp 3670ndash3678 1995

[13] L Kocarev and G Jakimoski ldquoLogistic map as a block encryp-tion algorithmrdquo Physics Letters A vol 289 no 4-5 pp 199ndash2062001

[14] N K Pareek V Patidar and K K Sud ldquoImage encryption usingchaotic logistic maprdquo Image and Vision Computing vol 24 no9 pp 926ndash934 2006

[15] A Kanso and N Smaoui ldquoLogistic chaotic maps for binarynumbers generationsrdquo Chaos Solitons amp Fractals vol 40 no5 pp 2557ndash2568 2009

[16] V Patidar K K Sud and N K Pareek ldquoA pseudo randombit generator based on chaotic logistic map and its statisticaltestingrdquo Informatica vol 33 no 4 pp 441ndash452 2009

[17] N Singh and A Sinha ldquoChaos-based secure communicationsystemusing logisticmaprdquoOptics and Lasers in Engineering vol48 no 3 pp 398ndash404 2010

[18] Y Liu H Fan E Y Xie G Cheng and C Li ldquoDeciphering animage cipher based on mixed transformed logistic mapsrdquo Inter-national Journal of Bifurcation and Chaos in Applied Sciencesand Engineering vol 25 no 13 Article ID 1550188 9 pages 2015

[19] D A Hsieh ldquoChaos and nonlinear dynamics application tofinancial marketsrdquo The Journal of Finance vol 46 no 5 pp1839ndash1877 1991

[20] N Basalto R Bellotti F De Carlo P Facchi and S PascazioldquoClustering stock market companies via chaotic map synchro-nizationrdquo Physica A Statistical Mechanics and Its Applicationsvol 345 no 1-2 pp 196ndash206 2005

[21] R Bowen and J Chazottes Equilibrium States and the ErgodicTheory of Anosov Diffeomorphisms vol 470 Springer 1975

[22] P H Borcherds and G P McCauley ldquoThe digital tent map andthe trapezoidal maprdquoChaos Solitons amp Fractals vol 3 no 4 pp451ndash466 1993

[23] T Addabbo M Alioto A Fort S Rocchi and V VignolildquoThe digital tent map performance analysis and optimizeddesign as a low-complexity source of pseudorandom bitsrdquo IEEETransactions on Instrumentation and Measurement vol 55 no5 pp 1451ndash1458 2006

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

Complexity 21

[24] R M Corless ldquoWhat good are numerical simulations ofchaotic dynamical systemsrdquo Computers amp Mathematics withApplications vol 28 no 10ndash12 pp 107ndash121 1994

[25] S Li G Chen and X Mou ldquoOn the dynamical degradation ofdigital piecewise linear chaotic mapsrdquo International Journal ofBifurcation and Chaos vol 15 no 10 pp 3119ndash3151 2005

[26] K J Persohn and R J Povinelli ldquoAnalyzing logistic mappseudorandom number generators for periodicity induced byfinite precision floating-point representationrdquoChaos SolitonsampFractals vol 45 no 3 pp 238ndash245 2012

[27] A S Mansingka A G Radwan M A Zidan and K NSalama ldquoAnalysis of bus width and delay on a fully digitalsignum nonlinearity chaotic oscillatorrdquo in Proceedings of the54th IEEE International Midwest Symposium on Circuits andSystems (MWSCAS rsquo11) pp 1ndash4 IEEE Seoul Republic of KoreaAugust 2011

[28] A S Mansingka M Affan Zidan M L Barakat A G Radwanand K N Salama ldquoFully digital jerk-based chaotic oscillatorsfor high throughput pseudo-random number generators up to877 Gbitssrdquo Microelectronics Journal vol 44 no 9 pp 744ndash752 2013

[29] I Ozturk and R Kilic ldquoCycle lengths and correlation propertiesof finite precision chaotic mapsrdquo International Journal of Bifur-cation and Chaos vol 24 no 9 Article ID 1450107 14 pages2014

[30] H Wang B Song Q Liu J Pan and Q Ding ldquoFPGA designand applicable analysis of discrete chaotic mapsrdquo InternationalJournal of Bifurcation and Chaos vol 24 no 4 Article ID1450054 2014

[31] E W Weisstein Logistic map From MathWorld-A WolframWeb Resource 2014 httpmathworldwolframcomLogistic-Maphtml

[32] M Perc ldquoUser friendly programs for nonlinear time seriesanalysisrdquo 2010 httpwwwmatjazperccomejptimehtml

[33] httpsenwikiquoteorgwikiMahatma Gandhi[34] A Rukhin J Soto J Nechvatal M Smid and E Barker ldquoA

statistical test suite for random and pseudorandom numbergenerators for cryptographic applicationsrdquo Tech Rep DTICDocument 2001

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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